As explained in Ref. [1]}, remarkably, the set of vectors \( \vec{V}_{J,s}\)  defines a cyclic polytope,
which for us simply means that the set of its boundaries can be written down explicitly very easily. They essentially take the form of determinants built out of pairs of consecutive vectors in the sum over spins. For example, for \(d=4\)  the relevant determinants take the form \((i , i+1 , j , j+1)\)  with \(j>i\) , where \(i\)  and \(j\)  label various spins in the sum (REF ).
